Problem: Table I contains outputs of the function $f(x)=b^x$ for some $x$ values, and Table II contains outputs of the function $g(x)=\log_b(x)$ for some $x$ values. In both functions, $b$ is the same positive constant. Fill in the missing values in the tables. If necessary, round your answer to three decimal places. You do not need a calculator. Table I $x$ $-1$ $1.585$ $2.585$ $4.17$ $f(x)=b^x$ $0.5$ $3$
Solution: The inverse relationship of exponents and logarithms By definition, we know that $f(x)=b^x$ and $g(x)=\log_b x$ are inverse functions. Therefore, if $(p,q)$ satisfies function $f$, then we know that $(q,p)$ must satisfy function $g$. Filling table I From the second table, we see that $(6,2.585)$ satisfies function $g$, and so $\log_b{6}=2.585$. This also implies that $b^{2.585}=6$, and so $(2.585,6)$ satisfies function $f$. [Why?] Filling table II From the first table, we see that $(1.585,3)$ satisfies function $f$, and so $b^{1.585}=3$. This also implies that $\log_b{3}=1.585$, and so $(3,1.585)$ satisfies function $g$. [Why?] Here are the complete tables: Table I $x$ $-1$ $1.585$ $2.585$ $4.17$ $f(x)=b^x$ $0.5$ $3$ ${6}$ $18$ Table II $x$ $0.5$ $2$ $3$ $6$ $g(x)=\log_{b}(x)$ $-1$ $1$ ${1.585}$ $2.585$